\[\boldsymbol{Y} = \begin{bmatrix} Y_1 \\ \vdots \\ Y_n \end{bmatrix} \sim \mathcal{GP}(\boldsymbol{\mu}, \boldsymbol{\Sigma})\]
where \(\boldsymbol{\mu} = \mu(t_i),\; \boldsymbol{\Sigma} = \mathrm{Cov}(Y_i, Y_j) = k(t_i, t_j)\quad i,j = 1, \dots, n\).
\[k(\tau; \lambda) = A \exp\left\{-\left( \frac{t_i - t_j}{\lambda}\right)^2 - \Gamma \sin^2 \left(\frac{\pi (t_i - t_j)}{P}\right) \right\} + \sigma_i^2 \delta_{ij}\]
where \(A\) is the amplitude, \(\lambda\) is the correlation timescale, \(\Gamma\) is a smoothness parameter, \(P\) is the rotation period, and \(\sigma\) is the white noise.
george package in Remcee package in RModel 1 vs Model 2 compared using BIC (lower is better) and Likelihood Ratio statistic.
Fit third model of RVs using above GP + Keplerian orbit
Examine the residuals after fitting to see any presence of a signal.
Can use GPs to jointly model H\(\alpha\) and RV simultaneously.
QP GP returns a rotational period of 124.7 days
No residuals signals suggesting no exoplanet
Very sparse data!
GP perhaps have overfit; can consider a simpler less flexible kernel.